In this section, we prove Theorem REF  which states that if \(\Sigma \)  does not
have a non-displaceable subsurface and \(\operatorname{FMap} (\Sigma )\)  does not have an essential shift then
\(\operatorname{Map} (\Sigma )\)  and \(\operatorname{FMap} (\Sigma )\)  are CB.
This implies that \(\operatorname{Map} (\Sigma )\)  and \(\operatorname{FMap} (\Sigma )\)  are quasi-isometric to a point and therefore do not have an interesting
geometry. To prove this theorem, we make use of the notion of an avenue surface first introduced in [1]}.
Recall that we always assume that \(\Sigma \)  is stable.
